Pure Public Goods : c A Numerical Example

Pure Public Goods : c A Numerical Example

Suppose that the utility function of person 1, as a function of her consumption x1 of a pure private good, and z1 of a pure public good, could be written

U 1(x1, z1) = 2 ln x1 + ln z1

(where "ln x1" means the natural logarithm of x1) and that the utility function of person 2 could be written (as a function of his consumption x2 of the private good and z2 of the public good)

U 2(x2, z2) = ln x2 + 2 ln z2 These are both examples of Cobb?Douglas utility functions 1 . The two people also have different preferences : relative to person 1, person 2 has a stronger taste for the public good and a weaker taste for the private good. Suppose that the equation of the economy's production possibility frontier 2 is

X = 120 - Z

In this example, the production possibility frontier is a straight line, with a (constant) slope of -1. The fact that the production possibility frontier 3 is a straight line means that the M RT -- which is just the slope of the PPF -- is a constant, which here equals 1.

With the preferences given above, the marginal rates of substitution of the two people, the ratios of the marginal utilities, are

M RS1

=

M Uz1 M Ux1

=

x1 2z1

M RS2

=

M Uz2 M Ux2

=

2x2 z2

Any efficient allocation must obey the these three conditions :

(1) everyone consumes the pure public good : z1 = z2 = Z (2) the allocation is on the production possibility frontier : x1 + x2 + Z = 120 (3) the Samuelson condition : M RS1 + M RS2 = 1 In this example, when condition (1) is used to substitute for z1 and z2, the Samuelson condition becomes

1 see, for example, the appendix to chapter 4 of Varian's Intermediate Microeconomics, 8th

edition 2 see, for example, chapter 32 of Varian's text 3 sometimes called the "production possibility curve

16

x1 + 2x2 = 1 2Z Z

(eg1)

Now any allocation (x1, x2, Z) which satisfies the Samuelson condition, and the feasibility constraint (2), is Pareto optimal.

For example, each of the following three allocations is an efficient allocation :

i : x1 = 0 ; x2 = 40 ; z1 = z2 = Z = 80 ii : x1 = 20 ; x2 = 30 ; z1 = z2 = 70 iii : x1 = 60 ; x2 = 10 ; z1 = z2 = 50 In other words, at least in this example, there is more than one efficient allocation, and more

than one efficient level of public good provision.

Equation (eg1) can be written

x1 + 4x2 = 2Z

(eg2)

Any allocation (x1, x2, Z), in which all the consumption levels are non?negative, and which satisfies the optimality condition (eg2) and the feasibility condition

x1 + x2 + Z = 120

(eg3)

will be efficient. Since equation (eg3) implies that x2 = 120 - x1 - Z, we can substitute for x2 in equation

(eg2) to get x1 + 4(120 - x1 - Z) = 2Z

or

Z = 80 - x1 2

(eg4)

In this example, equation (eg4) completely describes all the efficient allocations. Take any x1 0. Then calculate Z from equation (eg4), and then x2 from equation (eg3) : as long as all three numbers are non?negative, we have an efficient allocation.

So there are many efficient allocations in this example : each of them satisfies the Samuelson

condition (eg2) (as well as the feasibility condition (eg3). The fact that there are many efficient

allocations should not be surprising. Consider the problem of finding efficient allocations in an

economy with only private goods, as in AP/ECON 2350 (for example, as in the Edgeworth box diagram) 4. There will, in general, be many efficient allocations, some better for person #1, and

some better for person #2. Notice in this example, as we increase x1, we decrease Z,and decrease x2 as well. That is because, in this example, person #1 has the relatively weaker taste for the pure public good. If the welfare function 5 were to give more importance to person #1, how would we

4 as, for instance, in chapter 31 of Varian's text 5 as in chapter 33 of Varian's text

17

make her better off, and person #2 worse off? First of all, we would give more of the private good to person #1. But we would also choose to produce more of the pure private good : giving more weight to person #1 in the welfare function means deciding on a production plan for the economy which is closer to her preferred plan, and she wants more of the private good and less of the public good.

Aside : A Particular Welfare Function

Consider the maximization of the welfare function W (U 1, U 2) = aU 1 + U 2

where a is some positive constant. The higher is a, the more the welfare function gives importance

to person #1. In this case,

W U 1 W1 = a

W U 2 W2 = 1

If you go back to the derivation of the efficiency conditions in the previous note, the first?order

conditions for the social planner's optimization were

W1Ux1 =

W2Ux2 = W1Uz1 + W2Uz2 = -F (Z)

In this example

Ux1

=

2 x1

so that equation (x1) implies that

2a =

x1

Also

Uz1

=

1 Z

Uz2

=

2 Z

so that equation (Z) implies that

a2 + =

ZZ

Combining equations (swf 1) and (swf 2),

2 + a 2a

=

Z

x1

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(x1) (x2) (Z )

(swf 1)

(swf 2)

or 2+a

Z = 2a x1 Finally, plugging equation (swf 3) into the efficiency condition(eg4), we get

(swf 3)

2+a 2a x1

=

80

-

x1 2

or which then imply

a

x1

=

a

+

80 1

2+a

Z=

80

2(a + 1)

(swf 4) (swf 5)

and

40

x2 = a + 1

(swf 6)

Equations (swf 4), (swf 5), and (swf 6) define completely the welfare?maximizing solution. For

any positive level of a, the relative weight on person #1's well?being, these three equations define

a feasible allocation which satisfies the Samuelson condition. For any positive level of a, the

allocation defined by equations (swf 4), (swf 5) and (swf 6) is efficient. As the weight a on person

#1's well?being goes up.x1 increases, and x2 and Z decrease.

Left to the Reader

What would happen if person 1 had a stronger taste than person 2 for the public good, for example if U 1 = ln x1 + 3 ln z1 and U 2 = ln x2 + ln z2?

What would happen in both people had the same taste for the public good, for example if U 1 = 4 ln x1 + ln z1 and U 2 = 4 ln x2 + ln z2? Would there still many efficient allocations? Would there still be many efficient levels of public good provision?

( tricky ) Here's another example in which both people have the same taste for the public

good

--

but

preferences

are

not

Cobb?Douglas

:

U1

=

200 x1

+ z1

and

U2

=

200 x2

+ z2

Adding up the Demand Curves Vertically

Returning to the original example, in which

U 1(x1, z1) = 2 ln x1 + ln z1

U 2(x2, z2) = ln x2 + 2 ln z2 recall that the demand functions of a person with Cobb?Douglas preferences

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are 6

U (x, z) = a ln x + b ln z

xD = a M a + b Px

zD = b M a + b Pz

where Px and Pz are the prices of the goods, and M the person's income. Plugging into the example, the two people's demand curves for the pure public goods have the equations

z1D

=

1 3

M1 Pz

z2D

=

2 3

M2 Pz

( These demand curves confirm an allegation made at the beginning of this section, that person

2 has a stronger taste for the pure public good than does person 1. The demand curve for person

2 is above and to the right of the demand curve of person 1. )

To add up the demand curves vertically, these equations must be expressed in "inverse de-

mand" format, showing how much each person is willing to pay as a function of the quantity she

or he consumes of the pure public good.

Since each person will consume the same amount of the pure public good in an efficient

allocation ( if the good is indeed a "good" ), the quantity each person consumes is just Z, the

quantity provided of the pure public good. The height of each person's demand curve is the price

she or he is willing to pay for a little more of the pure public good, which might be different for different people. Let Pz1 denote how much person 1 is willing to pay for a little more of the public good, and Pz2 how much person 2 is willing to pay. Then the equations for the demand curves can be written as

Z

=

1 3

M1 Pz1

Z

=

2 3

M2 Pz2

which can be re?arranged to express each person's willingness to pay as a function of the

quantity provided of the public good :

6 see the appendix to chapter 5 of Varian's text

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